Exercise 1 (pts 5). Prove that Σσι(α) = ” + Οζω log(n). . - Τ. η<α...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
1. Prove that log2(n) is O(n) 2. Prove that log(n!) is O(n log(n))
We have a dataset with n = 10 pairs of observations (li, yi), and η Σ α: = 683, Σμι = 813, i=1 η i=1 η Σ? = 47, 405, Σαιξε = 56, 089, Συ - Συ? = 66, 731. i=1 What is an approximate 99% confidence interval for the mean response at xo = 90?
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and σ(n) is the sum of those divisors. 4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
e s. Eacn (part of a) problern = 10 pts. Nam 1. Prove that Vn2+1-n 2. Prove that for any a, b>0, we have a, b 1, loga n-e(log, n).
(5) For n = 5, Verify the following summation formula: Σ + 1 (1) (η + 2)2"-1 10
Exercise 7 (2 points) Recall the binomial coefficient for integer parameters 0 Sk< n. Prove that Exercise 8 (2 points) Prove the following: if z is an integer with at most three decimal digits aia2a3, then x is divisible by 3 if and only if aut a2 +a3 is divisible by 3. Exercise 9 (3 points) A square number is an integer that is the square of another integer. Let x and y be two integers, each of which can...
Recall that ?-n/ ??-1 log Xi is the mle of ? for a beta(8.1) distribution.Also -_ ? ial log Xi has the gamma distribution ?(n.18) (a) Show that 2eW has a x2 (2n) distribution (b) Using part (a), find c1 and c2 so that 2?? for 0 < ? < 1 . Next, obtain a (1-?)100% confidence interval for ? (c) For ? = 0.05 and n = 10, compare the length of this interval with the lengthof the interval...
Exercise 15: Let (cn) be a sequence of positive numbers. Prove: lim infºn+1 < lim infch/n. n700 Cnn +00 What is the corresponding inequality for the lim sup?
1. Prove that log(1+r) = geometric series. (-1)-1 for <1. The simplest way to do this is to integrate the